中南大学课件.pptVIP

Discrete LTI systems: the convolution sum Consider signal x[n]: An arbitrary sequence can be representedas a linear combination of shifted unit implulses ?[n-k], where the weights in this linear combination are x[k]. Write as : Discrete LTI systems: the convolution sum ( sifting property —筛选性 ) Discrete LTI systems: the convolution sum Let h[n] denote the response of linear system to ?[n]. i.e. h[n] — the unit impulse response. then, each ?[n] of x[n]?response: …. …. Discrete LTI systems: the convolution sum …. …. This result is referred to as the convolution sum, and the operation on the right-hand side of EQ. is known as the convolution sum of x[n] and h[n]. Discrete LTI systems: the convolution sum We represent the operation as y[n] =x[n] ? h[n] (2.7) The same, is referred to as the convolution sum of x[n] and ?[n]. Some notes: An LTI system is completely characterized by its h[n]. Discrete LTI systems: the convolution sum The graph of convolution sum. Transform independent variable: x[n], h[n]?x[k], h[k] and h[k] ? h[-k] Shift h[-k] n steps ?h[n-k]. For any n, x[k] multiplied by h[n-k] and ?x[k]·h[n-k] . Discrete LTI systems: the convolution sum Example 2.2 determine y[n]=x[n]?h[n]. Answer: (a) Discrete LTI systems: the convolution sum and h[k]?h[-k] (b) Shift h[-k] to the right(n0) or to the left (n0). n0, y[n]=0 n=0, y[0]=x[0]?h[0] =0.5?1=0.5 n=1, y[1]=x[0]h[1]+x[1]h[0] =0.5+2=2.5 Discrete LTI systems: the convolution sum (c) For any particular value of n, we multiply these two signals and sum over all values of k . n=2, y[2]=x[0]h[2]+x[1]h[1] =0.5+2=2.5 n=3, y[3]=x[1]h[2]=2 n?4, y[n]=0 Discrete LTI systems: the convolution sum or y[